A SHORT OVERVIEW OF THE PROBLEMS

There are three sections to the material, each is quite different from the others and each stands alone.

Modeling Differential Equations with Difference Quotients and Curve Fitting

This section is particularly good for students who are also taking AP Statistics, but the curve fitting techniques are developed as if the students haven't seen it before. We all recognize one of the great powers of differential equations lies in the simplicity of the equation compared to the function it describes. Think about the logistic DE compared to its solution as just a simple example.

We can use that idea in some powerful ways in data analysis. If we collect the data (x,y) we can create difference quotients and then perform our curve fitting on the more simple difference quotients, then solve using the standard techniques of the AP course.

The section begins with an explanation of the process and some examples. Then the technique is used for models of

The Effect of Poaching on Gorilla Populations
Water Waste Via Leaking Pipes
Carbon Dioxide in the Atmosphere

The final project is a great data collection project which investigates the effect of blubber in insulating seals. The seals are rubber gloves filled with warm water and the blubber is Crisco (a complete description of the activity is given in the teacher notes). Students will measure the temperature of the water in the glove under a variety of conditions (with blubber, without blubber) and they can also see the effect of appendages (fingers tied in knot, fingers extended) on the rate of heat loss.

Student Handout (or pdf)
Teacher Notes and Solutions

The Pharmacokinetics of Medications

Students may have noticed that Advil suggests that you take a 200 mg pill every 6 hours, but Tylenol has a 325 mg tablet every 6 hours, and Aleve a 220 mg tablet every 12 hours. How is the size of the tablet and interval determined? What happens when you take a medication daily (for example, a 10-day course of antibiotics)? Why do they often start the treatment with a special pill or shot? What happens when you take a tablet containing two different medications (antihistamine and decongestant, for example)? Is it an antihistamine with a decongestant thrown in or a decongestant with some antihistamine? It's one or the other since the optimal intervals rarely coincide.

This series of problems leads students through these ideas with some open-ended Euler's Method investigations which include the rate of absorption into the system and the metabolization of the medication and its long-term behavior.

There is a short problem on continuous infusion of glucose, which requires the technique of curve fitting from the first section. This can be skipped if the curve fitting problem is not chosen.

The final problem involves showing that an integral equation (called a convolution) is equivalent to a standard differential equation when the absorption of the medication happens over a period of time with a known distribution (this models nicely the effects of hormone secretions).

Student Handout (or pdf)
Teacher Notes and Solutions

Predator Prey Models

This module presents the classic predator-prey model using foxes and rabbits. It includes supporting Excel documents for Euler's method. By using the Excel documents, students can investigate the behaviors of the process and asks for explanations for the relationships between the critical values for predator and for prey. A link to a Netlogo applet simulation (using wolves and sheep) of the process can be used for comparison.

The module extends the standard by adding in logistic growth for the rabbits. Students can also consider the effects of standard methods for limiting overabundant populations. Suppose we have too many rabbits: what would happen if bait is put out containing a birth control agent for the rabbits which reduces the growth rate by half. What would happen if you had hunt to "cull the herd"? While this is silly for rabbits, but is a common method for deer and other animals. Suppose we add more of the natural predator? Students can see in this simple model how difficult it is to control population with just two interacting species and also, how important timing is in the growth cycle. What will work well if implemented when the population is large, can be disastrous when the population is moderate or small.

Student Handout (or pdf)
Teacher Notes and Solutions and Excel sheet

If you choose to have your students work through this investigation and find errors or have questions, please e-mail me at teague at ncssm dot edu. One of my colleagues or I will get back to you with support.

Note: Teacher handouts are password-protected. Please email teague@ncssm.edu for credentials.